For the differential geometers out there: "Given B, a compact subsetof R^n with non-empty interior is the condition that B is strictlyconvex equivalent to some other condition involving some sort ofcurvature on S the boundary of B?"

Depends on the meaning of "some sort" and "strictly convex".

I understand strictly convex to mean that a chord connectingany two points on the boundary has only its endpoints on the boundary and the rest in the interior of the set.

Consider the function: f(x) = \sum 2^{-n}|x-r_n| where {r_n : n=1,2,3,...} is an enumeration of the rationals. Then the supergraph {(x,y): y >= f(x)} is strictly convex, but f''(x)=0 almost everywhere.

However, f'(defined at every irrational) is strictly increasing, so "some sort" of curvature condition might be said to be satisfied, defined in terms of the turning of tangent lines.